Hierarchical Cooperation for Operator-Controlled Device-to-Device Communications: A Layered Coalitional Game Approach

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by  Xiao Lu

Hierarchical Cooperation for Operator-Controlled
Device-to-Device Communications: A Layered
Coalitional Game Approach
Xiao Lu, Ping Wang, Dusit Niyato
School of Computer Engineering, Nanyang Technological University, Singapore
Email: {Luxiao, Wangping, Dniyato}@ntu.edu.sg
Abstract—Device-to-Device (D2D) communications, which al- operators for multi-path routing [3]. The cooperation can
low direct communication among mobile devices, have been increase throughput for the devices because a cooperative
proposedasanenableroflocal servicesin3GPPLTE-Advanced
relay may substantially lead to improved network capacity.
5
(LTE-A)cellularnetworks.Thiswork investigatesahierarchical
1 Accordingly, a larger amount of user traffic demand can be
LTE-AnetworkframeworkconsistingofmultipleD2Doperators
0 at the upper layer and a group of devices at the lower layer. supported, which will lead to higher aggregated revenue for
2 We propose a cooperative model that allows the operators to operators. In this cooperation, each operator needs to decide

n improvetheirutilityintermsofrevenuebysharingtheirdevices, on which operators to cooperate with to maximize profit and,
a and the devices to improve their payoff in terms of end-to-end giventhecooperationbehaviorofoperators.Then,thedevices
J throughputbycollaborativelyperformingmulti-pathrouting.To
helpunderstandingtheinteractionamongoperatorsanddevices, from cooperative operators need to make decision on which
4
wepresentagame-theoreticframeworktomodelthecooperation devicestocooperatewithtomaximizethroughput.Wecallthe

behavior, and further, we propose a layered coalitional game formation of this interrelated operator cooperation and device
]
I (LCG) to address the decision making problems among them. cooperation as a hierarchical cooperation problem, which is
N
Specifically, the cooperation of operators is modeled as an
the main focus of this paper. The hierarchical cooperation
. overlappingcoalitionformationgame(CFG)inapartitionform,
s gives rise to two major concerns. Firstly, what is the stable
in which operators should form a stable coalitional structure.
c
[ Moreover, the cooperation of devices is modeled as a coalitional coalitionalstructure desirablefor all operatorsso thatnone of
graphical game (CGG), in which devices establish links among operators is willing to leave the coalition? Secondly, what is

1 each other to form a stable network structure for multi-path thestablenetworkstructureforcooperativedevicestoperform
v routing.Weadopttheextendedrecursivecore,andNashnetwork,
multi-path routing? This paper addresses these two concerns
0 as the stability concept for the proposed CFG and CGG,
by formulatinga layered game framework to model the LTE-
2 respectively. Numerical results demonstrate that the proposed
6 LCG yieldsnotable gainscompared to both thenon-cooperative Anetworkwithoperatorsanddevicesbeingtheplayersinthe
0 case and a LCG variant and achieves good convergence speed. upper and lower layer, respectively. Previous work has also
0 Keywords- D2D communications, LTE-Advanced network, considered game-theoretic framework with hierarchies/layers,
.
1 layered coalitional game, coalitional structure formation, e.g., [4] in the cognitive radio networks, [5] in two-tier
0 multi-pathrouting,extendedrecursivecore,coalitionalgraph- femtocell networks and [6] in WRNs. However, most of the
5 ical game, Nash network. worksconsideredthe competitionrelationshipbetween differ-
1
ent layers, which belongs to the Stackelberg game concept
v: I. INTRODUCTION [7]. In the proposedlayered game framework,differentlayers
Xi Device-to-Device (D2D) communications [1], [2] have interacttoimprovethebenefitofeachothercooperatively.We
emerged as a promising paradigm for 3GPP LTE-Advanced adopt the concepts of an extended recursive core and Nash
r
a (LTE-A) networks, which provide mobile wireless connectiv- network as the solutions for the proposed games in the upper
ity, reconfigurable architectures, as well as various wireless layerandlowerlayer,respective.Tothebestofourknowledge,
applications(e.g.,networkgaming,socialcontentsharingand this is the first work to introduce the application of extended
vehicular networking) for better user experience. With D2D recursive core in wireless communications.
communications, nearby devices in a cellular network can II. NETWORK MODELAND PROBLEM FORMULATION
communicate with each other directly bypassing the base
stations. Conventional D2D communications commonly refer We consider an LTE-A network consisting of a number of
to direct information exchanges among devices in Human- devices belonging to multiple operators. We denote the set
to-Human and Machine-to-Machinecommunications,without of operators as H = {1,2,...,H}, and the set of devices of
the involvementof wireless operators. However, conventional operator h ∈ H as M(h) = {1,2,...,M }. The operators
h
D2D technologies cannot provide efficient interference man- are willing to form overlapping coalitions to maximize their
agement,securitycontrolandquality-of-serviceguarantee[2]. individual profits. An overlapping coalitional structure for a
Recently, there is a trend towards operator-controlled D2D number of operators can be defined in a cover function as
communications to facilitate profit making for operators as the set π = {S ,S ,...,S } which is a collection of non-
H 1 2 z
well as better user experience for devices [2]. empty subsets of H such that z S =H,∀k,S ⊆H and
k=1 k k
This paper considers a multi-hop LTE-A network consists the sub-coalitionscould overlaSpwith each other. z is the total
of devices deployed by multiple operators. In this network, number of coalitions in collection π . Let Γ denote the set
H h
an efficient approach to improve the end-to-end throughput of coalitions that operator h belongs to.
is to enable cooperative sharing of idle devices among the For multiple access at everyhop,we consider an OFDMA-
session l(h), i.e., i6=s(l(h)) and i6=d(l(h)), then
j6=s(l(h)) p6=d(l(h))
f (l(h))= f (l(h)). (2)
(i,j) (p,i)
jX∈Ai pX∈Ai
Let c denote the capacity of link (i,j). The aggregated
ij
data rate on each link (i,j) cannotexceed the link’s capacity.
Thus we have the following constraint,
i6=d(l(h)),j6=s(l(h))
f (l(h))≤c . (3)
(i,j) ij
SX∈ΓhhX∈S l(h)X∈L(h)
Let f⋆ (l(h)) denote the maximal rate a flow session l(h)
Fig.1. SystemmodelofOperator controlled D2DCommunications (i,j)
that is available on link (i,j), with the constraints in (1), (2)
and (3), and F⋆(l(h)) the maximal aggregated rate of a flow
based transmission1. In an operator coalition S ⊆ H, each session l(h). We have F⋆(l(h)) = ij=∈sS(il(h))f(⋆i,j)(l(h)). The
relay devicei∈M(h) notonlyneedsto supportinternalflow aggregated rate of flow session l(h)P∈L(h) is constrained by
transmission demand, but also can serve as a relay for other
D(l(h)), D(l(h))≤F⋆(l(h)),
devices from cooperative operators h′ ∈ S \{h}. Due to the r(l(h))= (4)
(cid:26) F⋆(l(h)), Otherwise,
limitedtransmissionpowerofeachdevice,multi-hoprelaying
is adopted to route flow sessions from source devices to where D(l(h)) is the rate demand of flow session l(h).
destinationdevices.Since single path routingis toorestrictive
forsatisfyingtrafficdemand,weassumeeachflowsessioncan
III. LAYEREDCOALITIONAL GAME FORHIERARCHICAL
be split for multi-path routing if necessary.
LTE-A NETWORKS
Figure1illustratesanexampleforthestudiedsystemmodel
and the corresponding notations. In this example, the LTE-A A. Layered Coalitional Game Framework
network is composed of 8 devices deployed by 3 different
operators, i.e., H = {O1,O2,O3}, MO1 = {D1,D2,D3}, Weformulateagame-theoreticframework,referredtoasthe
MO2 = {D4,D5,D6} and MO3 = {D7,D8}. From Fig. layeredcoalitionalgame(LCG),tomodelthedecision-making
1, there is a multi-path flow sourced from D1 to D2 and processof hierarchicalcooperationbetween the operatorsand
a single-path flow sourced from D8 to D7. From the cho- devices. Both operators and devices are assumed to be self-
sen links, the final coalitional structure of the operators is interested and rational. The operators aim to maximize their
{(O1,O2),(O2,O3)}. individualutility, while the devices attempt to maximize their
Let L = {1,2,...,l(h)} denote the set of flow sessions end-to-end throughput with the help of relay devices from
h
fromoperatorh. N(l(h)) denote the set of nodesof flow l(h). cooperative operators. In this LCG, the operators need to
Thesourceanddestinationdeviceofflowl(h)isrepresentedas decide on the coalitional structure and the devices need to
s(l(h)) and d(l(h)), respectively. We denote the link between make decisions to form a relay network structure, both in
distributed ways, with the aim to improve their utilities and
two devices i and j as (i,j).
payoffsrespectively.As both operatorsand devices only have
The channel gain on a link (i,j) can be obtained from [9]:
limited information at their own layer, information exchange
g =β·d−n,whereβ istheantennarelatedconstant,nisthe
ij ij between both layersis required.The operatorsneed to collect
path loss exponent, and d is the distance between devices i
ij the payoff information from devices, to make the decision
and j.
of coalitional structure formation. The decision of operators
f (l(h)) denotes the data rate on link (i,j) attributed to
(i,j) will then be provided to the devices for the purpose of
a flow session l(h) ∈ L(h),h ∈ S ⊆ H. Since, for D2D
network structure formation. In this regard, there could be
communications, a flow session from a source device may
multiple interactions between the operator layer and device
traverse multiple relay devices to reach its destination device,
layer. Recognizing the behavior of operators and devices,
we consider the following two cases about a device.
we propose to use the overlapping coalition formation game
1) If a device i is the source or destination of flow session (CFG) to model the behavior of operators and the coalitional
l(h), i.e., i=s(l(h)) or i=d(l(h)), then graphical game (CGG) to characterize the interaction among
f (l(h))=r(l(h)) or f (l(h))=r(l(h)), (1) devices,whichwillbeintroducedinSectionIII-BandSection
(i,j) (p,i)
III-C, respectively.
jX∈Ai pX∈Ai
Since operators and devices have different objectives and
where r(l(h)) is the aggregated rate of flow session l(h), and concernsduringthe cooperation,ournextstep is to definethe
Ai denotes the set of devices having direct link with i. objectivefunctionsto capture the incentivesfor operatorsand
2) If a device i is an intermediate relay device of flow devices.ForagivenoperatorcoalitionS,wedefinethepayoff
of a device i ∈ M(h) from operator h ∈ S, which performs
flow transmission, as the end-to-end throughput of the flow
1Othermultipleaccesstechniques canbeusedwithoutlossofgenerality
intheanalysis ofthispaper from device i to device k, which is expressed as follows,
maximize their aggregated profit, i.e., revenue is subtracted
i=n(l(h)) bydeviceoperationcostandcoalitioncost.Giventhepartition
v({i})= f (l(h)) (5) πH, we define the utility of an operator coalition S ∈ πH as
(i,j)
the profit of the coalition as follows,
SX∈ΓhhX∈Sl(hX)∈Lh jX∈Ai
with the constraints in (2), (3), (4) i=s(l(h))
where n(l(h))∈N(L(h)). uπH(S)= f(i,j)(l(h))·PR
A relay device j aims to help a device i on transmission to hX∈Sl(h)X∈L(h) jX∈Ai
improve the throughput by most. Therefore, to evaluate how − PiC − ξh(S) (8)
much the relay device j can help to improve the throughput hX∈Sl(h)X∈L(h)i∈NX(l(h)) hX∈S
of the device i, we define the payoff of the relay device j as with the constraints in (2), (3), (4)
the differencebetweenthethroughputofthe devicei with the
B. Overlapping Coalition Formation Game
help of device j and that without the help of device j, which
can be expressed as follows: An overlappingCFG is formulatedamongoperatorswhose
interests are to satisfy its internal flows with as less cost
as possible. Due to interference, the utility of any operator
v({j})=v(j)({i})−v(/j)({i}) is affected by not only the behavior of others in the same
i=n(l(h)) coalition,butalsothatofoperatorsfromothercoalitions.Thus,
= f (l(h)) theconsideredoperatorcoalitionalgameis ina partitionform
(i,j)
SX∈ΓhhX∈Sl(hX)∈Lhj∈AXi∪{j} since the aggregated utility of a coalition S ∈ πH depends
on the coalitional structure π of all the operators H in the
i=n(l(h)) H
network.We introducethe frameworkof an overlappingCFG
− f (l(h)) (6)
(i,j) in partition form with non-transferable utility to model the
SX∈ΓhhX∈Sl(hX)∈Lh jX′∈Ai
cooperation among operators.
with the constraints in (2), (3), (4) Definition1. AnoverlappingCFGinpartitionformwithnon-
transferableutility(NTU)isdefinedbyapair(H,u)whereH
where v(j)({i}) represents the payoff of device i with the
isthesetofplayers,anduisavaluefunctionthatmapsevery
helpofdevicej andv(/j)({i})representsthepayoffofdevice
partition π and every coalition S ∈ H,S ∈ π to a real
iwithoutthehelpofdevicek.IntheproposedLCG,operators H H
number that represents the total utility (profit) that players in
are allowed to form overlapping coalitions to share their idle
coalition S can obtain.
devices as relays with the aim to improve the aggregated
throughput. The strategy of an operator is to form the coalitions to
Throughmulti-path routing, each operator aims to improve improveitsindividualutilitydefinedby(7).Notethatdifferent
theaggregatedthroughputasmuchaspossible.Thus,inreturn, from non-overlapping CFG where players have to cooperate
higher revenue for providing the flow to meet the customer withallothersinthesamecoalitionandeachplayeronlystays
demand will be rewarded for each operator. We define the in one coalition, in an overlapping CFG, each player is able
individual utility of the operator as the profit, i.e., revenue to join multiple different coalitions.
minus the cost of devices in transmitting and forwarding a The solution of the overlapping CFG is the stable overlap-
packet (e.g., due to energy consumption). Let P denote the ping coalitional structure for operators, under which no one
R
rewarded revenue per unit throughputachieved per unit time, will deviate. To this end, we adopt the concept of extended
and PC denote the operation cost per device i per unit time. recursive core [8], referred to as γ†-core, as the solution.
i
In this case, we assume P >> PC. Then, given a partition γ†-core is an extended solution of coalition formation game,
R i
π ,foranoperatorhwithoutcooperation,i.e.,{h}∈π ,the whichallowscoalitionstooverlap,accountingforexternalities
H H
utility function uπH({h}) can be calculated as (7), across coalitions. In the proposed game, the externalities are
representedbytheinter-coalitioninterferencebetweendevices.
i=s(l(h)) Deviation is a key notion for the definition of the γ†-core.
uπH({h})= f (l(h))·P
(i,j) R Asaconsequenceofdeviation,anewpartitionwillbeformed.
l(h)X∈L(h) jX∈Ai Therefore,thedeviationisequivalenttotheformationofanew
− PC. (7) partition. In the proposed operator coalitional game,
i
l(h)X∈L(h)i∈NX(l(h)) Definition 2. Let partition π move to π′ by deviation.
with the constraints in (2), (3), (4) H H
• Complete deviation: If there exists DC ⊆H and πDC ⊆
The first term and the second term on the right side of (7) π′ such that for all h ∈ DC, for all coalition (S,π)
representsthe aggregatedrevenue for operator h and the total such that h ∈ S,S ∈/ π , then the player set DC
DC
device costs, respectively. performs complete deviation. The players h ∈ DC are
While cooperation can lead to profit improvement for op- called complete deviators.
erators, it may also incur inherent coordination costs, such as • Partial deviation: If there exists DP ⊆ H containing
packet overhead. Let ξ(Sh) denote the coalition cost incurs to only overlapping players such that for all h ∈ DP, for
operator h for being coaliton S. The objective for operator all S ∈ π ,S ∈/ π′ , then the player set DP performs
H H
cooperation through cooperative sharing of devices, is to partial deviation. The players h∈DP are called partial
deviators. We propose the coalition formation algorithm for operators
in Algorithm 1 to reach the stable coalitional structures in
In an overlapping CFG in a partition form, if a coalition
γ†-core.
of players performs a complete deviation or partial deviation,
this may affect the payoffs of the remaining players. For
Algorithm 1 Distributed Coalition Formation Game
the remaining players, we then define the residual game as
Initial State
following:
In the starting network, the operators are partitioned by π =
H
Definition 3. Let (H,v) be an overlapping CFG in partition H=1,··· ,H with non-cooperative operators.
Coalition Formation Process
form.IfasubsetofplayersS hasalreadyorganizedthemselves
Phase 1 Network Discovery
into a certain partition π . A residual game (R,v) is an
S DevicesfromthesameoperatorsperformtheDynamicVirtual
overlapping CFG in a partition form defined on a set of
players R=H\S. The players in R are called residuals. Link Formation algorithm specified in Algorithm 2. Based
on the information feedback from device layer, each
The residual game is an overlappingCFG in partition form operator h calculates the corresponding utilityin
on its own. In the residual game, the players react to the non-cooperative case.
deviationonlyonthesetofremainingplayersincludingpartial Phase 2 Coalition Formation
Theoperators play their strategies sequentially in arandom
deviators which can play further deviation.
order.
Given two payoff vectors x,y ∈ R|H|, if ∀i ∈ S ⊂ H, repeat
xi >yi, and ∃j ∈S, xj >yj, we write x>S y. Let (x,πH) 1) Each operators h∈H sequentially engages in pair-
denote an outcome of the game, where x is a utility vector wise negotiations with another operator h′ ∈H\{h}
resultingfromapartitionπ .LetC(H,v)denotetheγ†-core to identify potential cooperator. During this process,
H
the devices from the operator pair perform Dynamic
ofgame(H,v), andΠ(H,v)denotethesetofallthepossible
Virtual Link Algorithm. Based on the information
partitionsofH.γ†-corecanbefoundinductivelyinfourmain
feedback, each operator calculates itspotential utility.
steps [8] 2) Based on the potential utility information and
1) Trivial Game: Given a coalitional game (H,v), the history, the pair of operators decides to form a new
γ†-core of a coalitional game with H = {1} is composed coalition S ={h,h′} if it ensures the utility
of the only outcome with the trivial partition: C({1},v) = improvement.
3)Theoperatorsalreadythathavecooperationwithany
(v({1});{1}).
2) Inductive Assumption: Given the γ†-core C(R,v) for operator in S update their utility. Based on the
each game with |R| <|H| players, the assumption about the updatedutilityandhistory,theyperformdeviationwith
residual game (R,v) is defined as follows: the operator(s) in S if it leads to utility improvement.
until any further coalition formation does not result in
C(R,v), C(R,v)6=∅, utility improvement of at least one operator, i.e., conver-
A(R,v)=(cid:26) Π(R,v), Otherwise, (9) gence to a stable partition in the γ†-core.
Phase 3 Cooperative Sharing
3) Dominance: An outcome (x,π ) is dominated via a The operators share their relay devices with cooperative
H
coalition S if for at least one (y ,π ) there exists operators for multi-path routing according to the final
an outcome ((y ,y ),π ∪ π H)\S∈ ΠH(\HS ,v) such that network graph G⋆F.
S H,\S S H\S
(y ,y )> x.
S H\S S
4) γ†-core Generation: The γ†-core of a game of |H| The convergence of the operator coalitional game is guar-
players is the set of undominated outcomes. anteed due to the fact that 1) the total number of possible
The concept of dominance expresses that, given a current partitionswithoverlappingcoalitionsisfinite,2)thetransition
partition π and the respective payoff vector x, an undomi- from a partition to another leads to the increase of individual
H
nated coalition S represents a deviation from π in such a utility, and 3) the game contains mechanism to prevent the
H
way that the reachedoutcome((y ,y ),π ) is more operators to re-visit a previously formed coalitional structure.
S H\S S∪πH\S
rewardingfortheplayersin coalitionS, comparedtox.Thus, Asaresult,eachcooperationbuildupandbreakupwillleadto
γ†-core can be seen as a set of partitions, under which the anewpartition.Asthenumberofthepartitioncanbevisitedis
playerscooperateinself-organizedoverlappingcoalitionsthat finite, the game is guaranteed to reach a final partition, under
provides them with the highest payoff. which the utility of each operator can be no longer increased.
During the coalition formation process of the operator Furthermore the last partition lies in the γ†-core because 1)
coalitionalgame,inordertoreachanoutcomeliesinγ†-core, during the coalition formation process only the partitions that
we let each operator iteratively joins and leaves the coali- bring improvement of individual utility for each operator are
tions to ensures a maximum payoff (i.e., an undominated formed, and 2) in the final convergent partition, there are no
outcome). To prevent loop, we introduce a variable couple dominated coalitions which the operators are better off by
history (H(Γ ),H(v({h}))) for each operator to record all deviating from.
h
the coalitions that it has ever joined and the corresponding
C. Coalitional Graphical Game
utility. If the new coalition for operator h to join has been
recordedandtheutilitythatitisabouttogetisthesameasthe In the CGG, the source devices need to play transmis-
history, then operatorh will maintain the currentcoalition set sion strategy, while the relay devices need to play not only
Γ even if its utility will better off. Once operator h changes transmissionstrategybutalsorelaystrategy.Thetransmission
h
its coalition set, the new coalition set is included in history. strategy for a source device i is to send a link establishment
proposal to a relay j ∈ M(h),h ∈ Γ which can help to Algorithm 2 Distributed Multi-path Routing
h
improve its payoff defined in (5). The relay strategy for a Initial State
relay device j is to accept or reject a link proposal from In the starting network, each source device transmits directly
to its destination device.
a transmitting device i ∈ M(h),h ∈ Γ which increases
h Network Structure Formation Process
its payoff defined in (6). Once the relay device accepts the
repeat
link establishment proposal, it will need to play transmission Phase 1 Dynamic Virtual Link Replacement
strategy like a source device. Thedevicesplaytheirstrategiessequentiallyinarandom
order.
We consider that an device i can have multiple incoming repeat
andoutgoingflowssimultaneously,constrainedby(2),(3)and 1)Duringeachiterationy,everydeviceontransmis-
(4), and the maximum transmit power is limited to Q⋆. We sion performs pairwise negotiation with other idle
i devices fromcooperative operators and calculatesits
denotes thestrategyspacewhichconsistsofallthestrategies
i potential payoff improvement under cooperation.
of device i. When device i plays strategy si, while all other 2) After negotiation, each device i plays its best
devices maintain their current strategies denoted by a vector response s⋆i, based on its flow rate demand.
s , the resulting network graph is denoted by G . until none of the devices can further improve its
−i si,s−i payoff by a unilateral change in its strategy.
All the devices are considered to be myopic in the sense Phase 2 Feedback
Each device i sends the information about the link
that each device responds to improve the payoff given the
current strategies of the other devices. That is, each devices (i,j)∈G⋆F back to itsoperator for a coalition
formation decision.
plays myopicallywithout the foresightof the future evolution untilAcceptance of the convergent network structure by all
of the network. Based on this, we define the concept of best the operators which perform Algorithm 1, i.e., Algorithm 1
response for devices as follows: converges.
Phase 3 Multi-hop Routing
All the source and relay devices perform multi-path
Definition 4. A strategy s⋆i ∈ Si is a best response for a routing according to the final network structure G⋆F.
device i∈N, if v{i}(Gs⋆,s−i)≥v{i}(Gsi,s−i), ∀si ∈Si.
Based on the concept of best response, we introduce the
IV. NUMERICAL RESULTS
myopic dynamics algorithm for the proposed CGG shown in
Algorithm 2. We define an iteration as a round of myopic A. Simulation Setting
plays during which each device chooses to play its current Forsimulation,weconsideranLTE-AnetworkwithaTDD-
best response s⋆i ∈ Si sequentially in a random order with OFDMAscheme,locatingwithina1000m×1000marea.The
the aim to maximize its payoff given the current strategies bandwidth available in this network is 2MB. The maximum
of others. The dynamic virtual link formation process may transmit power of each device is 20mW. We calculate the
consist of one or more iterations, as the best strategy of capacityofeachlinkaccordingtoShannoncapacity.Thenoise
each device may change over time. All the devices play best levelis−90dBm.Forthewirelesspropagation,asin[13],we
strategiesbasedontheirflowdemands.Eachsourcedevicehas set the path loss exponentn=4 and antenna related constant
a certain self-generated flow demand, while the flow demand β = 62.5. Four operators are considered in this LTE-A
of a relay deviceis equalto its incomingthroughput.To meet networks.Foreachoperator,thereisoneinternalflowsession.
the flow demand, the device on transmission may propose Foreachflowsession,thesourcedeviceanddestinationdevice
a link establishment proposal to multiple relay devices. The are randomly selected. The number of devices from each
iteration stops when either all the flow demands of devices operator is varied from 3 to 8. The results presented in this
are satisfied or none of the devices can unilaterally change section are averaged over 1000 times of run with random
its strategy to improve its payoff. In other words, when the location of devices.
algorithm converges, it results in a network in which none
of the devices can unilaterally improve its throughput. This B. ConvergenceBehavior of the CoalitionalGraphicalGame
is referred to as a Nash network, which gives the stability We first examine the convergence speed of the CGG at
concept of the final network structure G⋆F. Specifically, the the device layer. To this end, we set the coalition cost to be
Nash network is defined as follows. C = 0. In this case, the grand coalition is always one of
the stable coalitional structures. Hence, the simulation with
Definition 5. A network graph G(N,E) with N denotingthe this setting is equivalent to performing the CGG given the
setofallnotes,i.e.devices,andE denotingthesetofalledges, coalitional structure of grand coalition at the operator layer.
i.e, links between pairs of devices, is the Nash network in its Each flow is assigned with a random rate requirement within
strategy space Si, ∀i ∈ N, if no device i ∈ N can improve [10,20]kb/s.InFig.2,weshowtheaverageandthemaximum
its payoff by a unilateral change in its strategy si ∈Si. numberofiterationsrequiredtillconvergenceofthealgorithm
fromtheinitialnetworkstructure(i.e.,directtransmission),as
In the Nash network, all the links are chosen based on the a function of the number of devices. As expected, with the
bestresponsesofdevicesandarethusintheNashequilibrium. increase of the number of devices, more interactions among
In a network with finite number of devices, the final network devices is required for the CGG to converge.Observing from
structure G⋆ resulting from the proposed CGG is a Nash Fig. 2, we find that when the total number of devices varies
F
network. from12to32,theaverageandmaximumnumbersofiterations
Number of iterations versus number of devices overlappingCFG allowsmorefreedomincoalitionalstructure
14
formation for potential utility improvement. In addition, we
Average number of iterations
12 Maximum number of iterations can observe that the performance gap between LCG and the
mber of iterations108 LneCtwGorvka.riant increases with the number of devices in the
Average nu 6 The hierarchical cVoo.pCerOaNtioCnLUiSnIOLNTE-A networks is a
4 promisingsolutionforhighspeeddatatransmissionandwide-
2 area coverage. In this paper, we have presented a game-
15 20 25 30
Total number of devices theoretic framework to model the hierarchical cooperation
problem. Specifically, we have proposed a layered coalitional
Fig.2. Numberofiterations versustotalnumberofdevices.
game(LCG)tomodelthecooperationbehavioramongtheop-
eratorsanddevicesinthe differentlayersofLTE-A networks.
Aggragrated utility versus number of devices
6000 Theconceptofextendedrecursivecorehasbeenadvocatedas
LGC
5000 LCG−variant the solution of stable coalitional structures. We proposed an
Non−cooperative
overlapping coalition formation game for operators to find a
4000
Aggregated utility23000000 stghtraaabtplhbeieccnaoelafilgtisatimoalnelatlhhasetsrcuboceoteupnreerianltiitevrsoedionupcetehrdaetfoeorxrst.ednWedvheidicleers,eactoucrosfaoivlriemtiocotnhraeel
stable network structures for multi-path routing. Numerical
1000
results have shown that the proposed LCG yields notable
0
12 14 16 18 20 22 24 26 28 30 32 gains relative to both the non-cooperative case and a LCG
Total number of devices
variant.Thefutureworkwillcharacterizetheperformancegap
Fig.3. Aggregated utility versustotalnumberofdevices. betweentheproposedLCGandtheoptimalsolutionsobtained
by centralized approaches.
ACKNOWLEDGEMENTS
vary from 2.35 and 5 to 5.86 and 14, respectively. Thus, on
average,theconvergencespeedofthealgorithmissatisfactory. This work was supportedin part by SingaporeMOE Tier 1
(RG18/13 and RG33/12),
C. Case Study of Four-operator Coalition with Uniformly
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